Tiêu đề MSTE 19 Solutions.pdf ✅

19 Integral Calculus: Integration Solutions ▣ 1. Find the area bounded by y = sin x and y = cos x from x = π 4 to x = 5π 4 . [SOLUTION] A = ∫ (sin x − cos x)dx 5π 4 π 4 A = 2√2 units 2 ▣ 2. Find the area between the curves 2x 2 + 4x + y = 0 and y = 2x. [SOLUTION] Solve for the points of intersection: 2x 2 + 4x + y = 0 y = −2x 2 − 4x y = y −2x 2 − 4x = 2x −2x 2 − 6x = 0 −2x(x + 3) = 0 x = 0, −3
A = ∫ [(−2x 2 − 4x) − 2x]dx 0 −3 A = 9 units 2 ▣ 3. What is the perimeter of the curve r = 4(1 − sin θ)? [SOLUTION] dr dθ = −4 cos θ L = 2 ∫ √[4(1 − sin θ)] 2 + (−4 cos θ) 2 3π 2 π 2 dθ L = 32 units ▣ 4. Evaluate the double integral ∫ ∫ (x 2 + y 2 ) 2y 0 dx 2 1 dy.
[SOLUTION] ∫ ∫ (x 2 + y 2 )dx 2y 0 dy 2 1 = ∫ ( 1 3 x 3 + xy 2) | 0 2y dy 2 1 = ∫ ( 14 3 y 3) dy 2 1 = 7 6 y 4|1 2 = 35 2 ▣ 5. The figure below is composed of a square with side 20 m and four quarter circles arranged as such. Find the area of the shaded region. [SOLUTION] The equation of the circular arc, (x − 0) 2 + (y − 0) 2 = r 2 x 2 + y 2 = 202